Splitting partially commutative Lie algebras into direct sums

TL;DR

This paper proves that certain partially commutative Lie algebras can be decomposed into direct sums of subalgebras based on the connectivity of their defining graph's completion.

Contribution

It establishes a necessary and sufficient condition for splitting such Lie algebras into direct sums, linking algebraic structure to graph connectivity.

Findings

Partially commutative Lie algebras split if and only if the graph's completion is disconnected.

The result applies to metabelian and nilpotent cases.

Provides a graph-theoretic criterion for algebra decomposition.

Abstract

In this work, we prove that partially commutative, partially commutative metabelian, or partially commutative nilpotent Lie algebra splits into the direct sum of two subalgebras if and only if the completion of the defining graph of this algebra is not connected.